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Entropies of Random Transformations on a Non-compact Space

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Abstract

In this paper, we present several possible natural extensions of metric and topological entropies for random transformations on a non-compact space. In particular, Brin–Katok local entropy formula and Katok \(\delta \) entropy formula will be reformulated in the context of non-compact setting. Furthermore, a corresponding variational principle for non-compact systems is given.

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References

  1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114(2), 309–319 (1965)

    Article  MathSciNet  Google Scholar 

  2. Bogenschütz, T.: Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dyn. 1(1), 99–116 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Bogenschütz, T.: Equilibrium states for random dynamical systems. Ph.D. Thesis, Bremen University (1993)

  4. Bowen, R.: Topological entropy for non-compact sets. Trans. Am. Math. Soc. 184, 125–136 (1973)

    Article  Google Scholar 

  5. Brin, M., Katok, A.: On Local Entropy. Lecture Notes in Mathematics 1007, pp. 30–38. Springer, Berlin (1983)

    MATH  Google Scholar 

  6. Crauel, H.: Random Probability Measures on Polish Spaces. Habilitationsschrift. Universitat, Bremen (1995)

    MATH  Google Scholar 

  7. Crauel, H.: Random Probability Measures on Polish Spaces, Stochastics Mono, vol. 11. Taylor & Francis, London (2002)

    Book  Google Scholar 

  8. Handel, M., Kitchens, B.: Metrics and entropy for non-compact spaces. Israel J. Math. 91, 253–271 (1995)

    Article  MathSciNet  Google Scholar 

  9. Katok, A.: Fifty years of entropy in dynamics: 1958–2007. J. Mordern Dyn. 1(4), 545–596 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Katok, A., Hasselblatt, B.: An Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ Press, Cambridge (1995)

    Book  Google Scholar 

  11. Katok, A.: Lyapunov exponents, entroy and periodic orbits for diffeomorphisms. Inst. Hautes Etudes Sci. Publ. Math. 51, 137–173 (1980)

    Article  Google Scholar 

  12. Kifer, Y.: Ergodic Theory of Random Transformations. Birkhauser, Basel (1986)

    Book  Google Scholar 

  13. Kifer, Y.: On the topological pressure for random bundle transformations. Trans. Am. Math. Soc. Ser. 202(2), 197–214 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Kifer, Y., Liu, P.-D.: Random dynamical systems. In: Hasselblatt, B., Katok, A. (eds.) Handbook of Dynamical Systems, vol. 1B, pp. 379–499. Elsevier, Hoboken (2006)

    Google Scholar 

  15. Li, Z.-M., Shu, L.: The metric entropy of random dynamical systems in a banach space: ruelle inequality. Ergodic Theory Dyn. Syst. 34(2), 594–615 (2014)

    Article  MathSciNet  Google Scholar 

  16. Li, Z.-M., Ding, Z.-H.: Remarks on topological entropy of random dynamical systems. Qual. Theory Dyn. Syst. 17(3), 609–616 (2018)

    Article  MathSciNet  Google Scholar 

  17. Liu, P.-D., Qian, M.: Smooth Ergodic Theory of Random Dynamical Systems. Lecture Notes in Mathematics, vol. 1606. Springer, Berlin (1995)

    Book  Google Scholar 

  18. Ma, D.-K., Cai, B.: Topological entropy of proper map. Taiwan J. Math. 18, 1219–1241 (2014)

    Article  MathSciNet  Google Scholar 

  19. Patrao, M.: Entropy and its variational principle for non-compact metric spaces. Ergodic Theory Dyn. Syst. 30, 1529–1542 (2010)

    Article  MathSciNet  Google Scholar 

  20. Mihailescu, E., Urbanski, M.: Random countable iterated function systems with overlaps and applications. Adv. Math. 298, 726–758 (2016)

    Article  MathSciNet  Google Scholar 

  21. Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago (1997)

    Book  Google Scholar 

  22. Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)

    Book  Google Scholar 

  23. Zhu, Y.-J.: Two notes on measure-theoretic entropy of random dynamical systems. Acta Math. Sin. Engl. Ser. 25(6), 961–970 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author would like to thank Professor Jon Aaronson and the School of mathematical sciences of Tel Aviv University for hospitality during his visit there.

Funding

The authors are supported by National Natural Science Foundation of China (No. 11871394), Israel Science Foundation (No. 1289/17), Natural Science Foundation of Shaanxi Province (No. 2019JM-123) and Natural Science Foundation of Shaanxi Provincial Department of Education (No. 18JK0770).

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Authors and Affiliations

  1. School of Mathematics, Northwest University, Xi’an, 710127, People’s Republic of China

    Zhiming Li & Dingxuan Tang

  2. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Zhiming Li

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  1. Zhiming Li
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  2. Dingxuan Tang
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Correspondence to Zhiming Li.

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Li, Z., Tang, D. Entropies of Random Transformations on a Non-compact Space. Results Math 74, 120 (2019). https://doi.org/10.1007/s00025-019-1046-3

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